Problem 33 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ \left(6^{1 / 3}\right)^{6} $$

Step-by-Step Solution

Answer

The evaluated result of the expression \( (6^{1 / 3})^{6} \) is 36.

Step 1: Identify the appropriate rule

This problem requires the understanding of the laws of exponents, more specifically, the rule of raising a power to another power. The expression, \( (6^{1 / 3})^{6} \), represents this situation.

Step 2: Apply the rule

Following the rule \( (a^{m})^{n} = a^{mn} \), substitute \( a=6, m=1/3, n=6 \) into the rule: \The expression becomes \( 6^{(1 / 3)*6} \).

Step 3: Multiply the exponents

To continue the evaluation, multiply the numbers in the exponent: \( 1/3 * 6 = 2 \).

Step 4: Simplify the expression

Substitute the result back into the expression to have \( 6^{2} \).

Step 5: Find the final value

Finally, simplify \( 6^{2} = 36 \). This is the numerical value of the given expression.

Key Concepts

Exponentiation

Understanding exponentiation is key in mastering various algebraic concepts. So what is exponentiation? It's a mathematical operation, where a number, called the base, is multiplied by itself a certain number of times indicated by an exponent. For example, when we say (6^{1 / 3})^{6}, we are dealing with a base of 6, which is raised to the power of 1/3, and this result is then raised to the power of 6. Exponentiation allows us to express large numbers or perform calculations on numbers that are in a power form more succinctly. Understanding this concept will pave the way for simplifying expressions and solving complex problems with ease.

Simplifying Expressions

The process of simplifying expressions involves reducing a complex mathematical expression into its simplest form using algebraic rules. This makes the expression easier to understand and work with. In our example, the expression (6^{1 / 3})^{6} seems complicated at first, but by applying the laws of exponents, we can simplify it step by step. Once we apply the rule of raising a power to a power, we multiply the exponents together, which simplifies the expression significantly. This simplified version can then be more easily evaluated to find its numerical value. For students, mastering simplification is essential for success in algebra and beyond.

Numerical Value

Finding the numerical value of an expression means to calculate its exact, simplified number. It's what you obtain after all the simplifications and calculations have been performed. In the example given, once we simplify the expression (6^{1 / 3})^{6} according to the laws of exponents, we are left with 6^{2}. Simplifying further, we calculate the power of 2 for the base number 6, which gives us a numerical value of 36. This final step is satisfying as it transforms abstract algebraic concepts into a clear and concrete number, which can be particularly useful in real-world applications and further mathematical computations.

Algebraic Rules

Algebra is grounded in a set of algebraic rules that govern the operations and transformations we can perform on expressions. Among the most fundamental of these are the laws of exponents, which guide how to manipulate expressions with powers. In the example provided, we use the rule that lets us multiply exponents when a power is raised to another power (a^m)^n = a^{mn}. Mastery of these rules allows students to tackle algebraic problems methodically and confidently. It's the foundation on which more complex and advanced mathematical concepts are built, making them indispensable in the toolkit of any math student.